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Geometrical methods of nonlinear analysis

Author: Mark A Krasnosel'skij; Petr P Zabrejko
Publisher: Berlin : Springer, 1984.
Series: Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 263
Edition/Format:   Print book : EnglishView all editions and formats
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Mark A Krasnosel'skij; Petr P Zabrejko
ISBN: 3540129456 9783540129455 0387129456 9780387129457
OCLC Number: 239841379
Notes: Aus dem Russ. übers.
Description: XIX, 409 Seiten.
Contents: 1. Vector Fields in Finite Dimensional Spaces.- 1. Extensions of Vector Fields.- 1.1 Vector Fields.- 1.2 Extensions of Vector Fields.- 1.3 Sard's Theorem.- 1.4 Extensions With a Minimal Number of Singular Points.- 2. Homotopic Vector Fields * *.- 2.1 Deformations and Homotopic Fields.- 2.2 Homotopy Criteria.- 2.3 Homotopy Classes.- 3. Rotation of Vector Fields.- 3.1 The Main Properties of the Rotation.- 3.2 Rotation in One-Dimensional Space.- 3.3 Rotation in Two-Dimensional Spaces.- 3.4 Rotation in Dimensional Space (n > 2).- 4. Theorems on Singular Points.- 4.1 Existence of Singular Points.- 4.2 Change of Domain.- 4.3 Algebraic Count of Singular Points.- 4.4 Index of the Singular Point at Infinity.- 5. Hopf's Theorem.- 5.1 Homotopy Classification.- 5.2 Vector Fields Without Singularities.- 6. Linear Vector Fields.- 6.1 Linear Fields.- 6.2 The Brouwer-Bohl Theorem.- 6.3 Computing the Index of Singular Points for Linearized Fields.- 6.4 Asymptotically Linear Fields.- 7. Product Theorems.- 7.1 A Product Formula for the Index.- 7.2 The Product Formula for Rotations.- 7.3 The Leray-Schauder Lemma.- 7.4 Direct Sum of Vector Fields.- 7.5 A Theorem on Reducible Vector Fields.- 7.6 The Sum Formula for Rotations.- 8. Periodic and Odd Vector Fields.- 8.1 Periodic Mappings.- 8.2 The Congruence Theorem.- 8.3 A Special Case.- 8.4 Odd and Even Vector Fields.- 8.5 A General Theorem.- 8.6 Vector Fields Which are Symmetric With Respect to a Subspace.- 9. Special Coverings of Spheres.- 9.1 The Genus of a Set.- 9.2 Genus With Respect to Periodic Mappings.- 9.3 Deformations Along Great Circles.- 9.4 Proof of Theorem 9.3.- 10. Homogeneous Polynomials.- 10.1 Parity of Rotation.- 10.2 Homogeneous Fields in the Plane.- 10.3 Quadratic Fields.- 11. Smooth Vector Fields.- 11.1 Positive Vector Fields.- 11.2 Analytic Fields in Complex Spaces.- 11.3 Singular Points of Analytic Fields.- 12. Gradient Fields.- 12.1 The Index of a Nonsingular Potential.- 12.2 Quadratic Potentials.- 12.3 Odd and Even Potentials.- 12.4 Homogeneous Potentials.- 12.5 Coercive Potentials.- 13. Periodic and Bounded Solutions of Differential Equations.- 13.1 The Setting of the Problem.- 13.2 The Translation Operator.- 13.3 Guiding Functions.- 13.4 Existence of Periodic Solutions.- 13.5 The Index of Periodic Solutions.- 13.6 Regular Guiding Functions.- 13.7 Examples.- 13.8 Existence of Solutions Which are Bounded on the Whole Axis.- 13.9 General Boundary Value Problems.- 14. Construction of Guiding Functions.- 14.1 Homogeneous Systems.- 14.2 Planar Systems.- 14.3 Equations With Polynomial Right-Hand Side.- 14.4 Periodic and Bounded Solutions.- 15. The Index of a Singular Point of a Planar Vector Field.- 15.1 General Theorems.- 15.2 Simple Singular Rays.- 15.3 Multiple Singular Rays (Noncritical Case).- 15.4 Multiple Singular Rays (Critical Case).- 15.5 A Geometrical Scheme.- 15.6 The Order of Nondegeneracy.- 2. Completely Continuous Vector Fields.- 16. Continuous Fields in Infinite Dimensional Spaces.- 16.1 Continuous Deformations of Fields.- 16.2 Contractions of Spheres.- 16.3 Leray's Example.- 16.4 Kakutani's Example.- 16.5 Passing to Smaller Classes of Vector Fields.- 17. Completely Continuous Operators.- 17.1 Definitions.- 17.2 Spaces and Operators.- 17.3 Linear Integral Operators.- 17.4 Spectral Properties of Completely Continuous Linear Operators...- 17.5 The Frechet Derivative.- 17.6 Taylor's Formula.- 17.7 The Substitution Operator.- 17.8 The Hammerstein Operator.- 17.9 The General Nonlinear Integral Operator.- 17.10 Asymptotic Derivatives of Integral Operators.- 18. Finite Dimensional Approximations.- 18.1 Projections on Convex Sets.- 18.2 Remarks.- 18.3 Finite Dimensional Approximation of Operators.- 18.4 The Schauder Projections.- 18.5 Extensions of Completely Continuous Operators.- 18.6 Extensions With a Finite Number of Fixed Points.- 19. Homotopy for Completely Continuous Vector Fields.- 19.1 Definitions.- 19.2 Homotopy Criteria.- 19.3 A Remark on the Definition of Homotopy.- 19.4 Uniformly Nonsingular Homotopies.- 19.5 Finite Dimensional Approximations for Compact Deformations.- 19.6 A Special Criterion for Homotopy.- 20. Rotation of Completely Continuous Vector Fields.- 20.1 Rotation of Fields With Finite Dimensional Operators.- 20.2 The Definition in the General Case.- 20.3 Properties of Rotation.- 20.4 Existence of Singular Points.- 20.5 Generalization of Hopfs Theorem.- 20.6 Fields With Zero Rotation.- 21. Linear and Almost Linear Completely Continuous Fields.- 21.1 Rotation of Linear Fields.- 21.2 Asymptotically Linear Fields.- 21.3 One-sided Estimates.- 21.4 Computing the Index of a Regular Fixed Point.- 21.5 Fields With (Blt GBP2)-Quasilinear Operators.- 21.6 Coercive Differentiable Fields Ill.- 21.7 A Special Case.- 21.8 Odd Fields.- 22. Product of Rotations.- 22.1 The Product Formula for Indices.- 22.2 The Product Formula for Rotations of Vector Fields.- 22.3 Vector Fields on a Direct Sum of Subspaces.- 22.4 Homeomorphisms and Completely Continuous Fields.- 22.5 Homeomorphisms on Boundaries of Domains.- 23. Smooth Completely Continuous Vector Fields.- 23.1 Smale's Theorem.- 23.2 Positively Oriented Completely Continuous Vector Fields.- 23.3 Completely Continuous Analytic Vector Fields.- 24. Computing the Index of a Singular Point in Critical Cases...- 24.1 The Setting of the Problem and Notations.- 24.2 The Case of a Simple Critical Point...- 24.3 The General Case.- 24.4 Fields With Analytic Principal Part.- 24.5 The Simple Critical Case.- 24.6 The General Critical Case.- 24.7 The Method of Successive Substitutions.- 24.8 Another Reduction Principle.- 3. Principles of Relatedness.- 25. Invariance Principles for the Rotation.- 25.1 Domains With a Common Core.- 25.2 A Counter-Example.- 25.3 Linear Fields.- 25.4 Some Basic Lemmata.- 25.5 Invariance Principles..- 26. Composition of Operators.- 26.1 The Setting of the Problem.- 26.2 The Main Theorem.- 26.3 Generalizations.- 26.4 Equivalent Conditions.- 26.5 Choice of a New Norm.- 26.6 The Index of a Singular Point for Iterates of Operators.- 27. Transition to Equations in a Subspace.- 27.1 The Setting of the Problem.- 27.2 A Lemma on the Index.- 27.3 The Principle of Relatedness.- 28. Forced Vibrations.- 28.1 A Periodic Boundary Value Problem.- 28.2 Transformations of Related Equations.- 28.3 Integral Equations.- 28.4 Change of Space.- 28.5 The Translation Field and the Basic Theorem.- 28.6 Applications of Guiding Functions.- 28.7 Equations in Banach Spaces.- 29. Boundary Value Problems.- 29.1 Periodic Problems.- 29.2 Construction of Equivalent Integral Equations.- 29.3 Integro-Differential Equations.- 29.4 The Principle of Relatedness for Periodic Problems.- 29.5 A General Boundary Value Problem.- 29.6 A General Principle of Relatedness.- 29.7 Remarks.- 30. The Principle of Relatedness for Elliptic Equations.- 30.1 Linear Equations.- 30.2 Nonlinear Equations.- 30.3 Theorems on the Equality of Rotations.- 30.4 The Principle of Relatedness.- 30.5 Generalizations -.- 31. Vector Fields Involving Iterated Operators.- 31.1 A General Theorem.- 31.2 Passage to a Finite Dimensional Subspace.- 31.3 Passage to Smooth Mappings.- 31.4 Completion of the Proof.- 31.5 Consequences and Generalizations.- 4. Fields With Noncompact Operators.- 32. The Method of Partial Redefinition of Operators.- 32.1 Compactly Supported Operators.- 32.2 Measures of Noncompactness.- 32.3 Fields With Condensing Operators.- 32.4 Computing the Rotation: Examples.- 32.5 Limit Compact Operators.- 33. Fields With Positive Operators.- 33.1 Cones and Partial Orderings.- 33.2 Positive Linear Operators.- 33.3 Nonlinear Positive Operators.- 33.4 Rotation of Fields With Positive Operators.- 33.5 Computing the Rotation.- 33.6 Computing the Index of a Singular Point * * * *.- 33.7 A Perturbation Result.- 33.8 Remarks.- 34. The Method of Partial Inversion.- 34.1 Fields With Invertible Operators.- 34.2 Quasirotation for Partially Invertible Fields.- 34.3 Contractions.- 34.4 An Important Example.- 34.5 A Generalization.- 1 34.6 Rotation mod 2.- 34.7 A Lemma on Linear Fredholm Operators.- 34.8 Nonlinear Fredholm Mappings.- 35. Some Generalizations.- 35.1 Approximation Schemes.- 35.2 Frum-Ketkov's Theorem.- 35.3 Operators in Topological Vector Spaces.- 35.4 Other Articles.- 36. Multivalued Mappings.- 36.1 Upper Semicontinuous Mappings.- 36.2 Rotation for Fields With Multivalued Operators.- 36.3 Fixed Points.- 36.4 Generalizations.- 36.5 Mappings in Quotient Spaces.- 5. Solvability of Nonlinear Equations.- 37. Invariant Sets and Fixed Points.- 37.1 Generalized Contractions.- 37.2 Schauder's Principle.- 37.3 A Priori Estimates.- 37.4 Completely Continuous Perturbations of Contractions.- 37.5 Nonexpansive Operators.- 37.6 Construction of Invariant Sets ?.- 37.7 Remarks.- 38. Fixed Points of Monotone Operators.- 38.1 Monotone Operators.- 38.2 Monotonically Limit Compact Operators.- 38.3 An Example.- 39. Dissipative Operators.- 39.1 Dissipative Equations.- 39.2 Browder's Principle.- 39.3 Existence of Periodic Solutions.- 39.4 Fixed Points for Dissipative Operators.- 39.5 Existence of a A-Centre.- 40. Almost Linear Equations.- 40.1 Equations With Completely Continuous Operators.- 40.2 Equations With Smooth Operators.- 40.3 A Lemma 47. Nonzero Solutions of Parametrized Equations.- 47.1 The Setting of the Problem.- 47.2 A Lemma on Nonhomotopic Fields.- 47.3 Continuous Branches of Solutions.- 47.4 Operators With a Monotone Minorant.- 47.5 The Principle of Topological Closure '...- 47.6 Eigenvectors of Concave Operators.- 48. Connectivity Principles.- 48.1 Normally Solvable Equations.- 48.2 Strongly Smoothable Operators.- 48.3 Equations With Nonexpansive Operators.- 48.4 Equations With Differentiable Operators.- 48.5 The Structure of the Solution Funnel.- 48.6 Remarks.- 7. Construction of Solutions.- 49. The Method of Successive Approximation.- 49.1 Convergence.- 49.2 Equations With Concave Operators.- 49.3 Contractions for Comparable Elements.- 49.4 Equations With Invertible Positive Operators.- 50. Approximating Equations.- 50.1 The Setting of the Problem.- 50.2 The Method of Galerkin-Petrov.- 50.3 The Ljapunov-Cesari Method.- 50.4 Tonelli's Method.- 50.5 Numerical Quadrature Methods.- 51. Error Estimates.- 51.1 The Setting of the Problem...- 51.2 A General Scheme.- 51.3 Remarks.- 52. The Index of a Stable Solution.- 52.1 Successive Iterations and Stability...- 52.2 A Criterion for Lack of Asymptotic Stability.- 52.3 Continuous Deformations of Systems With Isolated Equilibrium State...- 52.4 The Index of a Ljapunov-Stable Equilibrium State.- 52.5 Balls With Handles.- 52.6 Remarks.- 8. Small Perturbations of Nonlinear Equations.- 53. Perturbations and Existence Theorems.- 53.1 The Principle of Nonzero Rotation.- 53.2 Weakly Coupled Equations.- 53.3 An Application of the Principles on Relatedness and Invariance of Rotation.- 53.4 On a Class of Boundary Value Problems.- 54. Perturbations of Isolated Solutions.- 54.1 Solutions With Nonzero Index.- 54.2 The General Implicit Function Theorem.- 54.3 The Bifurcation Equation.- 54.4 (n, m)-Rotation.- 54.5 Systems of Scalar Equations With a Surplus Number of Unknowns...- 54.6 Parametrized Equations.- 54.7 Equations With Completely Continuous Operators.- 55. Functionalizing the Parameter.- 55.1 A General Scheme.- 55.2 The Index of a Cycle.- 55.3 Computing the Index of a Cycle.- 55.4 The Theorem on Relatedness.- 55.5 Autonomous Equations With Time-Lag.- 55.6 Remarks.- 56. The Principle of Changing Index.- 56.1 Bifurcation Points.- 56.2 Necessary Conditions.- 56.3 Sufficient Criteria.- 56.4 Continuous Branches of Nonzero Solutions.- 56.5 Leading Nonlinearities.- 56.6 Critical Loads.- 56.7 Solutions of Large Norm.- 57. Stability of Critical Values.- 57.1 Eigenvectors of Gradient Operators.- 57.2 Even Functionals.- 57.3 Small Perturbations.- 57.4 Some Lemmata on Weakly Continuous Functionals.- 57.5 Sets of Finite Genus.- 57.6 Stability of Critical Values.- 57.7 Bifurcation Points for Equations With Gradient Operators.- References.- List of Symbols.
Series Title: Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 263
Responsibility: M.A. Krasnosel'skiǐ ; P.P. Zabreǐko.

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